# A characterization of n-dimensional hypersurfaces in ${R}^{n+1}$ with commuting curvature operators

Banach Center Publications (2005)

- Volume: 69, Issue: 1, page 205-209
- ISSN: 0137-6934

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topYulian T. Tsankov. "A characterization of n-dimensional hypersurfaces in $R^{n+1}$ with commuting curvature operators." Banach Center Publications 69.1 (2005): 205-209. <http://eudml.org/doc/282030>.

@article{YulianT2005,

abstract = {Let Mⁿ be a hypersurface in $R^\{n+1\}$. We prove that two classical Jacobi curvature operators $J_x$ and $J_y$ commute on Mⁿ, n > 2, for all orthonormal pairs (x,y) and for all points p ∈ M if and only if Mⁿ is a space of constant sectional curvature. Also we consider all hypersurfaces with n ≥ 4 satisfying the commutation relation $(K_\{x,y\} ∘ K_\{z,u\})(u) = (K_\{z,u\} ∘ K_\{x,y\})(u)$, where $K_\{x,y\}(u) = R(x,y,u)$, for all orthonormal tangent vectors x,y,z,w and for all points p ∈ M.},

author = {Yulian T. Tsankov},

journal = {Banach Center Publications},

keywords = {commutation relation; Jacobi operator; skew-symmetric operator},

language = {eng},

number = {1},

pages = {205-209},

title = {A characterization of n-dimensional hypersurfaces in $R^\{n+1\}$ with commuting curvature operators},

url = {http://eudml.org/doc/282030},

volume = {69},

year = {2005},

}

TY - JOUR

AU - Yulian T. Tsankov

TI - A characterization of n-dimensional hypersurfaces in $R^{n+1}$ with commuting curvature operators

JO - Banach Center Publications

PY - 2005

VL - 69

IS - 1

SP - 205

EP - 209

AB - Let Mⁿ be a hypersurface in $R^{n+1}$. We prove that two classical Jacobi curvature operators $J_x$ and $J_y$ commute on Mⁿ, n > 2, for all orthonormal pairs (x,y) and for all points p ∈ M if and only if Mⁿ is a space of constant sectional curvature. Also we consider all hypersurfaces with n ≥ 4 satisfying the commutation relation $(K_{x,y} ∘ K_{z,u})(u) = (K_{z,u} ∘ K_{x,y})(u)$, where $K_{x,y}(u) = R(x,y,u)$, for all orthonormal tangent vectors x,y,z,w and for all points p ∈ M.

LA - eng

KW - commutation relation; Jacobi operator; skew-symmetric operator

UR - http://eudml.org/doc/282030

ER -

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